Examples of such surfaces include the plane minus a Cantor set and the tripod surface, which has exactly three ends, all accumulated by genus. These surfaces have finite-invariance index 2 and 3, respectively. Since these surfaces have nondisplaceable subsurfaces, their mapping class groups are not CB by the work of Mann–Rafi [1]}. However, Durham–Fanoni-Vlamis showed that neither surface has a good curve graph [2]}.
